Does cheese ever really go bad as it's already mold?
Would it loose freshness every time it is melted and reshredded or does that help?
What is the best way to melt cheese and add food dye that doesn't burn to a pan? I would like to make pizza with colorful designs in it.
You've asked three questions...
1) Yes, it will go bad; not every cheese is made with mold(s) added.
Time and temperature cause living things to die. Your cheese is a living thing and it degrades over time.
2) I would not recommend melting and re-shredding as the amount butterfat within the cheese will diminish each time.
3) As long as the dye is food safe, you can add it to the shreds before melting or, paint the designs on after baking.
Using a whole-milk mozzarella will have additional butterfat and would therefore not stick as easily.